import numpy as np
S, K, r, v, q, T, n = 41.0, 40.0, 0.08, 0.30, 0.0, 1.0, 1.0
h = T / n
u = np.exp((r - q)*h + v*np.sqrt(h))
d = np.exp((r - q)*h - v*np.sqrt(h))
u, d(np.float64(1.4622845894342245), np.float64(0.8025187979624785))Binomial Option Pricing: Selected Topics
McDonald Chapter 11
DATA 5695/6695
DATA 5695/6695
Tyler J. Brough
2025-02-18
McDonald Chapter 11 Binomial Option Pricing: Selected Topics
Understanding Early Exercise
Understanding Early Exercise
- Options may be rationally exercised prior to expiration
- By exercising, the option holder
- Receives the stock and thus receives dividends
- Pays the strike price prior to expiration (this has an interest cost)
- Loses the insurance implicit in the call against the possibility that the stock price will be less than the strike price at expiration
Understanding Early Exercise
- If volatility is zero, the value of insurance is zero. Then, it is optimal to defer exercise as long as interest savings on the strike exceed dividends lost
\[ rK > \delta S \]
- Therefore, it is optimal to exercise when
\[ S > \frac{rK}{\delta} \]
NB: in the special case when \(r = \delta\) and \(\sigma = 0\), any ITM option should be exercised immediately
When volatility is positive, the implicit insurance has value that varies with time to expiration
Understanding Early Exercise (Cont’d)
- The following graph displays the price, above which early exercise is optimal for a 5-year call option with \(K =\$100\), \(r = 5\%\), and \(\delta = 5\%\)
Understanding Early Exercise (Cont’d)
- The following graph displays the price, below which early exercise is optimal for a 5-year put option with \(K = \$100\), \(r = 5\%\), and \(\delta = 5\%\)
Understanding Risk-Neutral Pricing
Understanding Risk-Neutral Pricing
A risk-neutral investor is indifferent between a sure thing and a risky bet with an expected payoff equal to the value of the sure thing
\(p^{\ast}\) is the risk-neutral probability that the stock price will go up
Understanding Risk-Neutral Pricing (Cont’d)
The option pricing formula can be said to price options as if investors are risk-neutral
Note that we are not assuming that investors are actually risk-neutral, and that risky assets are actually expected to earn the risk-free rate of return
- Just that a world of perfect synthetic replication and costless/frictionless arbitrage is isomorphic to such a world!
Pricing an Option Using Real Probabilities
Is option pricing consistent with standard discounted cash flow calculations?
Yes! However, discounted cash flow is not used in practice to price options
This is because it is necessary to compute the option price in order to compute the correct discount rate
Pricing an Option Using Real Probabilities (cont’d)
Suppose that the continuously compounded expected return on the stock is \(\alpha\) and that the stock does not pay dividends
If \(p\) is the true probability of the stock going up, \(p\) must be consistent with \(u, d\), and \(\alpha\)
\[ puS + (1-p)dS = e^{\alpha h} S \]
- Solving for \(p\) gives us
\[ p = \frac{e^{\alpha h} - d}{u - d} \]
Pricing an Option Using Real Probabilities (cont’d)
- Using \(p\), the actual expected payoff to the option one period hence is
\[ pC_{u} + (1-p)C_{d} = \frac{e^{\alpha h} - d}{u - d}C_{u} + \frac{u - e^{\alpha h}}{u - d}C_{d} \]
- At what rate do we discount this expected payoff?
- It is not correct to discount the option at the expected return on the stock, \(\alpha\), because the option is equivalent to a leveraged investment in the stock and hence is riskier than the stock
Pricing an Option Using Real Probabilities (cont’d)
Denote the appropriate per-period discount rate for the option as \(\gamma\)
Since an option is equivalent to holding a portfolio consisting of \(\Delta\) shares of stock and \(B\) bonds, the expected return on this portfolio is
\[ e^{\gamma h} = \frac{S \Delta}{S \Delta + B} e^{\alpha h} + \frac{B}{S \Delta + B} e^{r h} \]
- NB: this is like a WACC for the option contract.
Pricing an Option Using Real Probabilities (cont’d)
- We can now compute the option price as the expected option payoff, equation(11.5), discounted at the appropriate discount rate, given by equation (11.6). This gives
\[ C = e^{-\gamma h} \left[\frac{e^{\alpha h} - d}{u - d} C_{u} + \frac{u - e^{\alpha h}}{u - d} C_{d} \right] \]
Pricing an Option Using Real Probabilities (cont’d)
It turns out that this gives us the same option price as performing the risk-neutral calculation
Note that it does not matter whether we have the “correct” value of \(\alpha\) to start with
Any consistent pair of \(\alpha\) and \(\gamma\) will give the same option price
Risk-neutral pricing is valuable because setting \(\alpha = r\) results in the simplest pricing procedure.
One-Period Numerical Example
One-Period Numerical Example (Figure 11.3, p. 328)
Let \(\alpha = 15\%\) per annum (just pulled out of your finger!)
Let \(S = \$41.0\), \(K = \$40.0\), \(r = 8\%\) per annum, \(\sigma = 30\%\) per annum, \(\delta = 0\), and \(T = 1\) year
One-Period Example (Cont’d)
One-Period Example (Cont’d)
The “true” or “physical” probability is then:
The option payoffs at expiry are:
One-Period Example (Cont’d)
The expected option payoff at expiry under the physical density is:
Q: But how to get its present value? We cannot use \(r\)!
One-Period Example (Cont’d)
The option weighted-average cost of capital (or risky discount rate):
- \(\Delta\) and \(B\) do not depend upon \(\alpha\) or \(p\):
(np.float64(0.7376478738781428), np.float64(-22.404982402058376))So now we can compute the portfolio weights:
One-Period Example (Cont’d)
So with \(\gamma\) in hand, we can now take the present value:
NB: This is the same answer we got with both the no-arbitrage and risk-neutral forms of the model
The Binomial Tree and Lognormality
The Binomial Tree and Lognormality
The usefulness of the binomial pricing model hinges on the binomial tree providing a reasonable representation of the stock price distribution
The binomial tree approximates a lognormal distribution
The Random Walk Model
It is sometimes said that stock prices follow a random walk
Imagine that we flip a coin repeatedly
Let the random variable \(Y\) denote the outcome of the flip
If the coin lands displaying a head, \(Y = 1\); otherwise, \(Y = -1\)
If the probability of a head is \(50\%\), we say the coin is fair
After \(n\) flips, with the \(i^{\text{th}}\) flip denoted \(Y_{i}\), the cumulative total, \(Z_{n}\), is
\[ Z_{n} = \sum_{i=1}^{n} Y_{i} \]
- It turns out that the more times we flip, on average the farther we will move from where we start
The Random Walk Model (cont’d)
- We can represent the process followed by \(Z_{n}\) in term of the change in \(Z_{n}\)
\[ Z_{n} - Z_{n-1} = Y_{n} \\ \]
\[ \begin{align} \text{Heads:} \quad & Z_{n} - Z_{n-1} = +1 \\ \text{Tails:} \quad & Z_{n} - Z_{n-1} = -1 \end{align} \]
The Random Walk Model (cont’d)
- A random walk, where with heads, the change in \(Z\) is 1, and with tails, the change in \(Z\) is \(-1\)
Simulating a Binomial Random Walk Process
Simulating Binomial Prices
The Random Walk Model (cont’d)
The idea that asset prices should follow a random walk was articulated in Samuelson (1965)
In efficient markets, an asset price should reflect all available information. In response to new information the price is equally likely to move up or down, as with the coin flip
The price after a period of time is the initial price plus the cumulative up and down movements due to informational surprises
Modeling Stock Prices As a Random Walk
The above description of a random walk is not a satisfactory description of stock price movements. There are at least three problems with this model
If by chance we get enough cumulative down movements, the stock price will become negative
The magnitude of the move ($1) should depend upon how quickly the coin flips occur and the level of the stock price
The stock, on average, should have a positive return. The random walk model taken literally does not permit this
The binomial model is a variant of the random walk model that solves all of these problems at once
The Binomial Model
\[ S_{t+h} = S_{t} e^{(r - \delta)h \pm \sigma\sqrt{h}} \]
- Taking logs, we obtain
\[ \ln\left(S_{t+h} / S_{t}\right) = (r - \delta)h \pm \sigma\sqrt{h} \quad\quad \text{(11.11)} \]
Since \(\ln\left(S_{t+k} / S_{t}\right)\) is the continuously compounded return from \(t\) to \(t+h\), the binomial model is simply a particular way to model the continuously compounded return
That return has two parts:
- The drift: \((r - \delta) h\) generates the upward trend (all else equal)
- The diffusion: \(\sigma\sqrt{h}\)) which is uncertain and generates the up and down moves
The Binomial Model (cont’d)
Equation (11.11) solves the three problems in the random walk
The stock price cannot become negative
As \(h\) gets smaller, up and down moves get smaller
There is a \((r-\delta) h\) term, and we can choose the probability of an up move, so we can guarantee that the expected change in the stock price is positive
Lognormality and the Binomial Model
The binomial tree approximates a lognormal distribution, which is commonly used to model stock prices
The lognormal distribution is the probability distribution that arises from the assumption that continuously compounded returns on the stock are normally distributed
With the lognormal distribution, the stock price is positive, and the distribution is skewed to the right, that is, there is a chance that extremely high stock prices will occur
Lognormality and the Binomial Model (cont’d)
- The binomial model implicitly assigns probabilities to the various nodes.
Lognormality and the Binomial Model (cont’d)
- The following graph compares the probability distribution for a 25-period binomial tree with the corresponding lognormal distribution
Is the Binomial Model Realistic?
- The binomial model is a form of the random walk model, adapted to modeling stock prices. The lognormal random walk model in this section assumes among other things, that
- Volatility is constant
- “Large” stock price movements do not occur
- Returns are independent over time
- All of these assumptions appear to be violated in the data